Sheffield United striker David McGoldrick scored his first international goal for the Republic of Ireland.
The attacker cancelled out a goal from Fabian Schar, which had given Switzerland the lead.
The intense European qualifier from Group D finished 1-1 at the Aviva Stadium.
It is a vital point for Ireland as they look to progress from a difficult group.
The Republic are currently top with 11 points from 5 games, but Denmark have 8 and a game in hand.